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Differentiation under the integral sign and holomorphy. (English) Zbl 0613.46043

A general theory of holomorphic functions between convergence vector spaces, with \(L_ c\)-embedded target spaces, was developed in [S. Bjon and M. Lindström, Monatsh. math. 101, 11-26 (1986; Zbl 0579.46031)] using continuous convergence on function spaces. There are, however, important classes of convergence vector spaces, which are not \(L_ c\)-embedded (e.g. non-Schwartz convex bornological vector spaces). A theory, which uses \(L_ e\)-embedded target spaces (i.e. spaces F such that the natural mapping \(F\to L_ eL_ eF\) is an embedding) and ”local uniform convergence” (denoted by \({}_ e)\) instead of continuous convergence on function spaces, is therefore developed in this paper. The class of these spaces is large enough to contain all Hausdorff locally convex spaces and all polar bornological vector spaces. A notion of continuous differentiability, called (D2), is shown to combine well with a definite integral, which is defined using duality theory. It leads e.g. to theorems on differentiation under the integral sign and to theorems on the continuity of the operation of integration. The results are applied to holomorphic functions with values in \(L_ e\)-embedded spaces. It is shown that a function is holomorphic (i.e. Gâteaux-holomorphic and continuous) iff it is complex differentiable in the sense (D2). Power series expansions, which converge in function spaces with the structure \({}_ e\), are obtained and a new general exponential law \(H_ e(U\times V,G)\cong H_ e(U,H_ e(V,G))\) is derived for spaces of holomorphic functions, where G is sequentially complete and \(L_ e\)-embedded. This exponential law contains as special cases those concerning hypoanalytic and Gâteaux-holomorphic functions (with compact convergence and finite- dimensional compact convergence respectively) [cf. M. Schottenloher, Analyse Fonct. Appl., C. R. Colloq. d’Analyse, Rio de Janeiro 1972, 261-270 (1975; Zbl 0308.46044)], and that concerning spaces of holomorphic functions on bornological vector spaces [cf. J. F. Colombeau, Differential Calculus and Holomorphy (1982; Zbl 0506.46001)].

MSC:

46G20 Infinite-dimensional holomorphy
46A99 Topological linear spaces and related structures
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
46G05 Derivatives of functions in infinite-dimensional spaces
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